Soft Comput (2017) 21:4953–4961
Logic of approximate entailment in quasimetric and in metric
Published online: 17 June 2016
© The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract It is known that a quasimetric space can be repre-
sented by means of a metric space; the points of the former
space become closed subsets of the latter one, and the role
of the quasimetric is assumed by the Hausdorff quasidis-
tance. In this paper, we show that, in a slightly more special
context, a sharpened version of this representation theorem
holds. Namely, we assume a quasimetric to fulﬁl separabil-
ity in the original sense due to Wilson. Then any quasimetric
space can be represented by means of a metric space such
that distinct points are assigned disjoint closed subsets.
This result is tailored to the solution of an open problem
from the area of approximate reasoning. Following the lines
of E. Ruspini’s work, the Logic of Approximate Entailment
(LAE) is based on a graded version of the classical entailment
relation. We present a proof calculus for LAE and show its
completeness with regard to ﬁnite theories.
A quasimetric is deﬁned similarly to a metric; the assumption
of symmetry, however, is dropped. This generalisation of the
concept of a distance naturally occurs in many real-world
situations. An often cited example is the time that a walker
needs to get from one place to another one within a mountain-
ous area. Quasimetric spaces are moreover closely related to
weighted directed graphs. The latter play a signiﬁcant role in
Communicated by A. Di Nola.
Department of Knowledge-Based Mathematical Systems,
Johannes Kepler University Linz, Altenberger Straße 69,
4040 Linz, Austria
computer science, in particular for the formulation of net-
work ﬂow problems (Ahuja et al. 1993; Heineman et al.
It is certainly also true that the notion of a quasimetric
is by far less common in mathematics than its symmetric
counterpart. Remarkably, however, the metric spaces them-
selves give rise to a non-symmetric distance function. The
Hausdorff quasidistance quantiﬁes the difference between
subsets of a metric space, rather than between its points. It
shares with a metric certain characteristic properties like the
triangle inequality, but symmetry does in general not hold.
Hyperspaces consisting of subsets of a metric space, endowed
with the Hausdorff quasidistance, are used in a number of
contexts. An example is the area of point-free geometry (Con-
cilio and Gerla 2006; Concilio 2013). Measuring the degree
of distinctness between subsets of a metric space is moreover
an issue in fuzzy set theory; see, for example, Gerla (2004).
It is reasonable to ask whether distance functions that vio-
late symmetry but otherwise resemble a metric always arise
from a metric space in the above way mentioned. We ﬁnd
the afﬁrmative answer in Vitolo’s paper (1995). Vitolo has
studied spaces (W, q), where q : W × W → R
triangle inequality as well as the following version of the sep-
aration axiom: q(a, b) = q(b, a) = 0 if and only if a = b.
He proved that any such space can be embedded into the
hyperspace of non-empty closed sets of a metric space, the
role of the quasimetric being taken by the Hausdorff quasi-
distance. Also among the results in Gerla (2004), Vitolo’s
representation theorem can be found.
In the present paper, we study distance functions of a more
special type. In this paper, a quasimetric is understood to be
a mapping q : W × W →
such that, for a, b, c ∈ W ,
q(a, c) q(a, b) + q(b, c), and q(a, b) = 0 if and only if
a = b. This deﬁnition is in accordance with the early work of
Wilson on the topic (1931); however, it is evidently stronger